Finite Group Databasebeta

← #29835482

Subgroups — #29835482

Order 15616.

826 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840204C211yesmaximal, characteristic, complemented
2C2×#16840199C211yesmaximal, characteristic, complemented
3C2×#16840195C211yesmaximal, characteristic, complemented
4C2×SD32×D61C211yesmaximal, characteristic, complemented
5#21456307C211yesmaximal, characteristic, complemented
6#21456301C211yesmaximal, characteristic, complemented
7#21448087C211yesmaximal, characteristic, complemented
8#21448085C211yesmaximal, characteristic, complemented
9#21448084C211yesmaximal, characteristic, complemented
10#21448081C211yesmaximal, characteristic, complemented
11#21448078C211yesmaximal, characteristic, complemented
12#21448076C211yesmaximal, characteristic, complemented
13#21448073C211yesmaximal, characteristic, complemented
14#21447417C211yesmaximal, characteristic, complemented
15C61×#2251C211yesmaximal, characteristic, complemented
16C2×#14519809—21no—
17C2×#14519804—21no—
18C2×#14519800C2211yescharacteristic
19C2×#14519800C2211yescharacteristic
20C2×Q16×D61C2211yescharacteristic
21C2×D8×D61C2211yescharacteristic
22#16840337—21no—
23#16840330—21no—
24#16840204—24no—
25#16840199—24no—
26#16840195—24no—
27C2×#14519723C2211yescharacteristic, complemented
28C2×#14519722C2211yescharacteristic
29C2×#14519722C2211yescharacteristic, complemented
30C2×#14519721C2211yescharacteristic
31C2×#14519720C2211yescharacteristic
32C2×#14519720C2211yescharacteristic
33C2×#14519720C2211yescharacteristic, complemented
34C2×#14519720C2211yescharacteristic, complemented
35C2×#14519720C2211yescharacteristic, complemented
36SD32×D61—24no—
37C2×D976C2211yescharacteristic
38C2×D976C2211yescharacteristic, complemented
39C2×#14519718C2211yescharacteristic
40C2×#14519716C2211yescharacteristic
41C2×C16×D61C2211yescharacteristic
42#16840155—21no—
43#16840154—21no—
44#16840152—21no—
45#16840148—21no—
46#16840147C2211yescharacteristic
47#16840147C2211yescharacteristic, complemented
48#16840146C2211yescharacteristic, complemented
49#16840140C2211yescharacteristic, complemented
50#16840139C2211yescharacteristic
51#16840137C2211yescharacteristic
52#16839716C2211yescharacteristic
53#16839537C2211yescharacteristic, complemented
54C4×#11758185C2211yescharacteristic, complemented
55C16×Dic61C2211yescharacteristic, complemented
56C2×Q32×C61C2211yescharacteristic, complemented
57C2×Q32×C61C2211yescharacteristic, complemented
58C2×SD32×C61C2211yescharacteristic
59C2×SD32×C61C2211yescharacteristic
60C61×#495C2211yescharacteristic
61C61×#494C2211yescharacteristic, complemented
62C61×#346C2211yescharacteristic, complemented
63C2×#11758226—21no—
64C2×#11758226—21no—
65C2×Q8×D61—21no—
66C2×D4×D61—21no—
67#14519809—44no—
68#14519804—44no—
69#14519800—22no—
70#14519800—22no—
71#14519800—22no—
72#14519800—22no—
73C2×Dic244C2311yescharacteristic
74C2×#11758202—21no—
75C2×#11758202—21no—
76Q16×D61—22no—
77Q16×D61—22no—
78C2×#11758190—21no—
79C2×#11758201—21no—
80C2×#11758201—21no—
81C2×#11758201—21no—
82C2×#11758201—21no—
83C2×D488—21no—
84C2×D488C2311yescharacteristic
85C2×D488C2311yescharacteristic
86C2×D488C2311yescharacteristic
87C2×#11758199—21no—
88C2×#11758199—21no—
89D8×D61—22no—
90D8×D61—22no—
91C2×#11758189D411yescharacteristic
92C2×C8×D61C2311yescharacteristic
93#14519775—21no—
94#14519774—21no—
95#14519773—21no—
96#14519766—21no—
97#14519766—21no—
98#14519765—21no—
99#14519762—42no—
100#14519755—42no—
101#14519750—42no—
102#14519747—42no—
103#14519734D411yescharacteristic
104#14519723—22no—
105#14519722—22no—
106#14519722—22no—
107#14519721—22no—
108#14519720—22no—
109#14519720—22no—
110#14519720—22no—
111#14519720—22no—
112#14519720—22no—
113D976—22no—
114D976—22no—
115#14519718—22no—
116#14519716D414yescomplemented
117C2×#11758185C2311yescharacteristic
118C2×#11758185C2311yescharacteristic
119C16×D61D414yescomplemented
120#14519688D411yescharacteristic
121#14519687D411yescharacteristic
122C8×Dic61C2311yescharacteristic
123C2×Q16×C61—21no—
124C2×Q16×C61C2311yescharacteristic
125C2×Q16×C61C2311yescharacteristic
126C2×Q16×C61C2311yescharacteristic
127C2×SD16×C61—21no—
128C2×D8×C61C2311yescharacteristic
129C4⋊Q8×C61—21no—
130#124×C61—21no—
131Q32×C61—22no—
132Q32×C61—22no—
133SD32×C61—22no—
134SD32×C61—22no—
135C2×C16×C61C2311yescharacteristic
136C2×C16×C61C2311yescharacteristic
137C4×C8×C61C2311yescharacteristic
138C23×D61—41no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×D4×C61—21no—
143C2×C22⋊D61—41no—
144C2×C22⋊D61—42no—
145#11758226—42no—
146#11758226—42no—
147#11758226—42no—
148#11758226—42no—
149Q8×D61—42no—
150Q8×D61—42no—
151D4×D61—42no—
152D4×D61—42no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—41no—
157C2×D244—41no—
158C2×D244—42no—
159C2×C4×D61—41no—
160C2×C4×D61—41no—
161C2×C4×D61—41no—
162C2×C4×D61C2×D411yescharacteristic
163C2×Dic122—21no—
164C2×Dic122—41no—
165Q16×C61—21no—
166Q16×C61—21no—
167Q16×C61—21no—
168Q16×C61—21no—
169Q16×C61—21no—
170Q16×C61—21no—
171Q16×C61—42no—
172SD16×C61—42no—
173D8×C61—21no—
174D8×C61—21no—
175C2×C8×C61C2×D411yescharacteristic
176C2×C8×C61C2411yescharacteristic
177C4⋊C4×C61—42no—
178C22⋊C4×C61—42no—
179C42×C61C2×D411yescharacteristic
180#11758202—42no—
181#11758202—42no—
182#11758201—42no—
183#11758201—42no—
184#11758201—42no—
185#11758201—42no—
186#11758199—42no—
187#11758199—42no—
188#11758198—42no—
189#11758198—42no—
190#11758198—42no—
191#11758198D812yes—
192#11758196—42no—
193#11758196D812yes—
194C4×Dic61C2×D411yescharacteristic
195C2×#92418C2×D411yescharacteristic
196C2×#92418C2×D411yescharacteristic
197Dic244—21no—
198Dic244—21no—
199D488—21no—
200D488—21no—
201D488—21no—
202D488—21no—
203D488—21no—
204D488—21no—
205D488—42no—
206#11758190—42no—
207#11758189D814yes—
208C8×D61C2×D412yes—
209C8×D61C2×D412yes—
210C16×C61C2×D412yes—
211C16×C61C2×D412yes—
212#11758185C2×D412yes—
213#11758185C2×D412yes—
214C23×C61—41no—
215C22×D61—41no—
216C22×D61—41no—
217C22×D61—41no—
218C22×D61—82no—
219C22×D61—162no—
220C22×D61C2×D811yescharacteristic
221Q8×C61—41no—
222Q8×C61—41no—
223Q8×C61—41no—
224Q8×C61—41no—
225Q8×C61—41no—
226Q8×C61—41no—
227D4×C61—41no—
228D4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61C2×D812yes—
233C2×C4×C61C22×D411yescharacteristic
234C22⋊D61—82no—
235C22⋊D61—162no—
236C2×Dic61—41no—
237C2×Dic61C2×D812yes—
238C2×Dic61C2×D811yescharacteristic
239D244—41no—
240D244—41no—
241D244—41no—
242D244—41no—
243D244—41no—
244D244—41no—
245D244—82no—
246D244—82no—
247D244—162no—
248C4×D61—82no—
249C4×D61—82no—
250C4×D61—82no—
251C4×D61#12712yes—
252C4×D61C2×D812yes—
253Dic122—41no—
254Dic122—41no—
255Dic122—82no—
256C8×C61C2×D812yes—
257C8×C61C22×D411yescharacteristic
258C8×C61C22×D411yescharacteristic
259#92418#12712yes—
260#92418C2×D812yes—
261#33722—611nomaximal, Sylow
262C22×C61—81no—
263C22×C61—82no—
264C22×C61C22×D811yescharacteristic
265C2×D61—22no—
266C2×D61—81no—
267C2×D61—81no—
268C2×D61—81no—
269C2×D61—82no—
270C2×D61—82no—
271C2×D61—82no—
272C2×D61—164no—
273C2×D61#49312yes—
274C4×C61—22no—
275C4×C61—81no—
276C4×C61—81no—
277C4×C61—81no—
278C4×C61C2×#12711yescharacteristic
279C4×C61C22×D811yescharacteristic
280Dic61—22no—
281Dic61—81no—
282Dic61#49312yes—
283C2×#510—611no—
284C2×#509—611no—
285C2×#508—611no—
286C22×SD32—611no—
287#3156—611no—
288#3155—611no—
289#2251—611no—
290#2251D6111yescharacteristic, complemented
291#2250—611no—
292#2250—611no—
293#2249—611no—
294#2249—611no—
295#2248—611no—
296#2248—611no—
297#2108—611no—
298C2×C61—161no—
299C2×C61C2×#49311yescharacteristic
300C2×C61#341612yes—
301D61—22no—
302D61—161no—
303D61—161no—
304D61—161no—
305C2×#137—1221no—
306C2×AGL(1,ℤ/8)—1221no—
307C2×#135—611no—
308C2×#135—611no—
309C22×Q16—611no—
310C22×D8—611no—
311#536—1221no—
312#535—1221no—
313#510—1224no—
314#509—1224no—
315#508—1224no—
316C2×Q32—611no—
317C2×Q32C2×D6111yescharacteristic, complemented
318C2×Q32C2×D6111yescharacteristic, complemented
319C2×SD32—611no—
320C2×SD32—611no—
321C2×SD32—611no—
322C2×SD32—611no—
323C2×SD32—611no—
324C2×SD32—611no—
325C2×SD32—611no—
326C2×SD32—611no—
327C2×SD32—1224no—
328C2×SD32C2×D6111yescharacteristic
329C2×SD32C2×D6111yescharacteristic
330C2×D16—611no—
331C2×D16—611no—
332C2×D16—611no—
333C2×M32—611no—
334C22×C16—611no—
335#497—1221no—
336#497—1221no—
337#496—1221no—
338#496—1221no—
339#495—611no—
340#495—611no—
341#495—611no—
342#495C2×D6111yescharacteristic
343#494—611no—
344#494C2×D6111yescharacteristic, complemented
345#493—611no—
346#493—611no—
347#405—611no—
348#346—611no—
349#346C2×D6111yescharacteristic, complemented
350C4×C16—611no—
351C4×C16—611no—
352C61#3372211yescharacteristic, complemented, Sylow
353C2×#41—1221no—
354C2×#41—1221no—
355C22×Q8—1221no—
356C22×D4—1221no—
357#137—2444no—
358AGL(1,ℤ/8)—2444no—
359#135—1222no—
360#135—1222no—
361#135—1222no—
362#135—1222no—
363C2×Q16—21no—
364C2×Q16—611no—
365C2×Q16—1222no—
366C2×Q16—1222no—
367C2×Q16—1221no—
368C2×Q16—1221no—
369C2×Q16C22×D6111yescharacteristic
370C2×Q16C22×D6111yescharacteristic
371C2×Q16C22×D6111yescharacteristic
372C2×SD16—21no—
373C2×SD16—1221no—
374C2×SD16—1221no—
375C2×SD16—1221no—
376C2×SD16—1221no—
377C2×SD16—1221no—
378C2×D8—611no—
379C2×D8—611no—
380C2×D8—611no—
381C2×D8—1222no—
382C2×D8—1222no—
383C2×D8—1221no—
384C2×D8—1221no—
385C2×D8—1221no—
386C2×D8C22×D6111yescharacteristic
387C2×M16—611no—
388C22×C8—611no—
389C4⋊Q8—21no—
390C4⋊Q8—1221no—
391#127—1221no—
392#127—1221no—
393#124—21no—
394#124—1221no—
395#124—1221no—
396#124—1221no—
397#123—2442no—
398C22⋊Q8—2442no—
399#121—2442no—
400#120—2442no—
401#117—611no—
402Q32—22no—
403Q32—22no—
404Q32—1222no—
405SD32—22no—
406SD32—22no—
407SD32—1222no—
408SD32—1222no—
409SD32—1222no—
410SD32—1222no—
411SD32—1222no—
412SD32—1222no—
413SD32—1222no—
414SD32—1222no—
415D16—1222no—
416D16—1222no—
417D16—1222no—
418M32—614no—
419C2×C16—611no—
420C2×C16—611no—
421C2×C16—614no—
422C2×C16C22×D6111yescharacteristic
423C2×C16C22×D6111yescharacteristic
424#97—611no—
425#97—611no—
426C4×C8—611no—
427C4×C8C22×D6111yescharacteristic
428C24—2441no—
429#41—2442no—
430#41—2442no—
431#41—2442no—
432#41—2442no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—1221no—
437C2×Q8—2441no—
438C2×Q8—2442no—
439C2×Q8—2442no—
440C2×D4—21no—
441C2×D4—1221no—
442C2×D4—1221no—
443C2×D4—1221no—
444C2×D4—2441no—
445C2×D4—2441no—
446C2×D4—2442no—
447C2×D4—2442no—
448C2×D4—2442no—
449C2×D4—2441no—
450C2×D4—2442no—
451C22×C4—611no—
452C22×C4—2441no—
453C22×C4—2441no—
454C22×C4—2441no—
455Q16—21no—
456Q16—21no—
457Q16—21no—
458Q16—21no—
459Q16—21no—
460Q16—21no—
461Q16—42no—
462Q16—1221no—
463Q16—1221no—
464Q16—2442no—
465Q16—2442no—
466SD16—42no—
467SD16—2442no—
468SD16—2442no—
469SD16—2442no—
470SD16—2442no—
471SD16—2442no—
472D8—21no—
473D8—21no—
474D8—1221no—
475D8—1221no—
476D8—1221no—
477D8—1221no—
478D8—1221no—
479D8—1221no—
480D8—2442no—
481D8—2442no—
482D8—2442no—
483M16—614no—
484C2×C8—611no—
485C2×C8—611no—
486C2×C8—612no—
487C2×C8—612no—
488C2×C8D4×D6111yescharacteristic
489C2×C8C23×D6111yescharacteristic
490C4⋊C4—42no—
491C4⋊C4—612no—
492C4⋊C4—2442no—
493C22⋊C4—42no—
494C22⋊C4—612no—
495C22⋊C4—2442no—
496C22⋊C4—2442no—
497C22⋊C4—2442no—
498C42—611no—
499C42D4×D6111yescharacteristic
500C16—612no—
501C16—612no—
502C16D4×D6112yes—
503C16D4×D6112yes—
504C23—41no—
505C23—611no—
506C23—9762no—
507C23—4882no—
508C23—2441no—
509C23—2441no—
510C23—2441no—
511Q8—41no—
512Q8—41no—
513Q8—41no—
514Q8—41no—
515Q8—41no—
516Q8—41no—
517Q8—4882no—
518Q8—2441no—
519Q8—2441no—
520D4—41no—
521D4—41no—
522D4—9762no—
523D4—9762no—
524D4—4882no—
525D4—4882no—
526D4—4882no—
527D4—2441no—
528D4—2441no—
529D4—2441no—
530D4—2441no—
531D4—2441no—
532D4—2441no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—611no—
537C2×C4—612no—
538C2×C4—612no—
539C2×C4—612no—
540C2×C4—4882no—
541C2×C4—4882no—
542C2×C4—4882no—
543C2×C4—2441no—
544C2×C4D8×D6112yes—
545C2×C4C2×D4×D6111yescharacteristic
546C8—612no—
547C8—612no—
548C8D8×D6112yes—
549C8C2×D4×D6111yescharacteristic
550C8C2×D4×D6111yescharacteristic
551C22—82no—
552C22—81no—
553C22—612no—
554C22—1222no—
555C22—9764no—
556C22—4882no—
557C22—4882no—
558C22—4882no—
559C22—4881no—
560C22—4881no—
561C22—4881no—
562C22C2×D8×D6111yescharacteristic
563C4—22no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4—612no—
568C4—1222no—
569C4—4881no—
570C4#127×D6111yescharacteristic
571C4C2×D8×D6111yescharacteristic
572C2—161no—
573C2—1222no—
574C2—9761no—
575C2—9761no—
576C2—9761no—
577C2#493×D6111yescharacteristic
578C2#2145896412yes—
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2251C2
socleC22×C61C22×D8
hypercenter#2251D61
hypocenterC61#33722
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2251

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16840204C2semidirect
2. C2×#16840199C2semidirect
3. C2×#16840195C2semidirect
4. C2×SD32×D61C2semidirect
5. #21456307C2semidirect
6. #21456301C2semidirect
7. #21448087C2semidirect
8. #21448085C2semidirect
9. #21448084C2semidirect
10. #21448081C2semidirect
11. #21448078C2semidirect
12. #21448076C2semidirect
13. #21448073C2semidirect
14. #21447417C2semidirect
15. C61×#2251C2semidirect
27. C2×#14519723C22semidirect
29. C2×#14519722C22semidirect
33. C2×#14519720C22semidirect
34. C2×#14519720C22semidirect
35. C2×#14519720C22semidirect
38. C2×D976C22semidirect
47. #16840147C22semidirect
48. #16840146C22semidirect
49. #16840140C22semidirect
53. #16839537C22semidirect
54. C4×#11758185C22semidirect
55. C16×Dic61C22semidirect
56. C2×Q32×C61C22semidirect
57. C2×Q32×C61C22semidirect
61. C61×#494C22semidirect
62. C61×#346C22semidirect
116. #14519716D4semidirect
119. C16×D61D4semidirect
290. #2251D61semidirect
317. C2×Q32C2×D61semidirect
318. C2×Q32C2×D61semidirect
344. #494C2×D61semidirect
349. #346C2×D61semidirect
352. C61#33722semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168402042
C2×#168401992
C2×#168401952
C2×SD32×D612
#214563072
#214563012
#214480872
#214480852
#214480842
#214480812
#214480782
#214480762
#214480732
#214474172
C61×#22512
#3372261